Find the length of the chord made by the axis of x, with the circle whose centre is (0,3) and which touches the straight line 3x + 4y = 37.

Answer

To find the length of the chord made by the axis of x with the circle whose center is (0,3) and which touches the straight line 3x + 4y = 37, we can follow these steps:

### Step 1: Identify the center and radius of the circle
The center of the circle is given as \( O(0, 3) \). The line \( 3x + 4y = 37 \) is a tangent to the circle.

### Step 2: Find the distance from the center of the circle to the tangent line
To find the radius of the circle, we need to calculate the perpendicular distance from the center \( O(0, 3) \) to the line \( 3x + 4y - 37 = 0 \) using the formula for the distance from a point to a line:
\[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 3, B = 4, C = -37 \), and the point \( (x_1, y_1) = (0, 3) \).

Substituting the values:
\[ \text{Distance} = \frac{|3(0) + 4(3) - 37|}{\sqrt{3^2 + 4^2}} = \frac{|0 + 12 - 37|}{\sqrt{9 + 16}} = \frac{|12 - 37|}{\sqrt{25}} = \frac{|-25|}{5} = 5 \]
Thus, the radius \( r \) of the circle is 5 units.

### Step 3: Determine the coordinates of the point where the radius meets the tangent line
The radius from the center to the tangent line is perpendicular to the line. Since the center is at \( (0, 3) \) and the radius is 5 units, we can find the coordinates of the point of tangency \( D \) using the direction ratios of the normal to the line. The slope of the line \( 3x + 4y = 37 \) can be found by rewriting it in slope-intercept form:
\[ 4y = -3x + 37 \implies y = -\frac{3}{4}x + \frac{37}{4} \]
The slope of the line is \( -\frac{3}{4} \), so the slope of the normal (which is the radius) is \( \frac{4}{3} \).

Using the point-slope form of the line equation:
\[ y - 3 = \frac{4}{3}(x - 0) \implies y = \frac{4}{3}x + 3 \]
To find the intersection of this line with the tangent line \( 3x + 4y = 37 \), substitute \( y \) into the tangent line equation:
\[ 3x + 4\left(\frac{4}{3}x + 3\right) = 37 \]
Simplifying:
\[ 3x + \frac{16}{3}x + 12 = 37 \implies \left(3 + \frac{16}{3}\right)x = 37 - 12 \]
\[ \frac{9}{3} + \frac{16}{3} = \frac{25}{3} \implies \frac{25}{3}x = 25 \implies x = 3 \]
Substituting \( x = 3 \) back to find \( y \):
\[ y = \frac{4}{3}(3) + 3 = 4 + 3 = 7 \]
Thus, the point of tangency \( D \) is \( (3, 7) \).

### Step 4: Find the length of the chord AB
Since the chord AB is bisected by the radius OC, we can find the length of half the chord \( BC \) using the Pythagorean theorem in triangle \( OBC \):
\[ OB^2 = OC^2 + BC^2 \]
where \( OB \) is the radius (5), and \( OC \) is the vertical distance from the center to the x-axis (3):
\[ 5^2 = 3^2 + BC^2 \implies 25 = 9 + BC^2 \implies BC^2 = 16 \implies BC = 4 \]
Since \( AB \) is twice \( BC \):
\[ AB = 2 \times BC = 2 \times 4 = 8 \]

### Final Answer
The length of the chord AB is \( 8 \) units.

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